Exercises: Proportional sharing

Divide the amount of $450$ € in direct proportion to the numbers $2$, $3$ and $4$.

Divide the number $300$ into parts inversely proportional to $2$ and $3$.

In a distribution directly proportional to the quantities $3$, $5$ and $7$, the first person receives $120$ €. Calculate the total amount shared and what the other two people receive.

Divide $330$ points in inverse proportion to $1$, $2$ and $3$.

Share $620$ shares in direct proportion to the fractions $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{5}$.

An amount of $130$ € is divided in inverse proportion to the numbers $2$, $3$ and $4$. How much corresponds to the part associated with the number $4$?

A grandfather decides to divide $120$ € among his three grandchildren aged $8$, $10$ and $12$ in direct proportion to their ages. Determine the amount of money each receives.

In a school competition, $190$ bonus points are awarded in inverse proportion to the number of mistakes made. If the three finalists made $1$, $3$ and $4$ mistakes respectively, calculate the points obtained by each student.

Hugo, Sara, and Leo have received $2$, $3$, and $6$ tardiness marks during the school term. To reward their end-of-year commitment, the tutor decides to distribute $720$ conduct points inversely proportional to the infractions committed by each of them.

Hugo proposes the following: «Since it is an inverse distribution, the person with the fewest infractions should get the highest coefficient of the group, and the one with the most, the lowest; therefore, the distribution coefficients should be $6$ for me, $3$ for Sara, and $2$ for Leo. We add $6 + 3 + 2 = 11$, calculate the constant, and distribute».

1. a) Prove with reasoning why the coefficient assignment proposed by Hugo is conceptually incorrect (compare the ratio between Hugo's and Sara's shares according to his proposal with the actual inverse ratio). Explain what typical mistake he made.
2. b) Correct the approach: write the correct coefficients, calculate the proportionality constant, and determine how many points each student receives. Check that the sum of the parts equals the total.

Three university students, Álex, Blanca, and Carlos, share a flat in Valencia and receive an electricity and gas bill of $210$ € for the month of July. Since they have spent weeks away, they agree to pay in direct proportion to the days each of them stayed in the apartment: Álex was there for $10$ days, Blanca for $20$ days, and Carlos was there the whole month ($30$ days).

1. a) Before performing any numerical calculation:

• Estimate whether Carlos will pay more, less, or exactly half of the total bill. Justify your estimate by analyzing the fraction of days Carlos stayed relative to the total sum of recorded days.

• Deduce with reasoning what exact relationship will exist between the amount Álex will pay and the amount Blanca will pay.

1. b) Carry out the distribution by calculating the direct proportionality constant $k$ and the exact amount assigned to each flatmate.
2. c) Check whether the obtained amounts confirm your estimates from part a) and verify the validity of the distribution by adding the three parts.

Three partners, Nerea, Beatriz and Santiago, invest $15000$ €, $20000$ € and $25000$ € respectively to open a business. After one year, the company generates a net profit of $14400$ €. Calculate how much of the profit each partner receives if it is shared in direct proportion to the capital invested.

In an athletics competition, the organisers share out a prize of $3400$ € in inverse proportion to the time taken by the top three finishers to cross the finish line. If they crossed the finish line in $12$, $15$ and $20$ minutes respectively, determine the cash prize awarded to each runner.

Four workers paint a building. The first works $5$ days at a rate of $6$ hours per day; the second, $4$ days at $8$ hours per day; the third, $6$ days at $5$ hours per day and the fourth, $3$ days at $10$ hours per day. If they earn a total of $3680$ € and decide to share it in direct proportion to the total hours worked by each, find how much each worker will receive.

An amount of money has been divided in inverse proportion to the ages of three siblings, who are $10$, $15$ and $20$ years old. If it is known that the $15$-year-old sibling received $400$ €, find the total amount shared and what the other two siblings receive.

An inheritance is divided in direct proportion to the numbers $4$, $7$ and $9$. Knowing that the heir who receives the largest share gets $3500$ € more than the one who receives the smallest share, calculate the total value of the inheritance and the amount assigned to each of the three shares.

In a laboratory experiment, a $145$-gram sample of a chemical compound must be divided among three containers in inverse proportion to their temperatures. The temperatures are $\frac{1}{2}$ ºC, $\frac{2}{3}$ ºC and $\frac{3}{4}$ ºC. Calculate the exact amount of compound that will be placed in each container.

To reforest a natural park, three municipalities provide different groups of volunteers. Municipality A provides $45$ volunteers; B, $60$ volunteers; and C, $75$ volunteers. The regional government awards them a total grant of $10800$ € for purchasing equipment, to be shared in direct proportion to the number of volunteers provided. What amount does each municipality receive?

A productivity bonus of $2530$ € is shared among three factory employees. The distribution is carried out in direct proportion to their years of service in the company ($4$, $5$ and $8$ years) and in inverse proportion to the number of days absent during the year ($2$, $3$ and $4$ days). Calculate the bonus received by each employee.

A municipal grant of $4550$ € for school sustainability projects is distributed among three schools inversely proportional to the time (in hours) that their automated system takes to sort a batch of waste: School A takes $\dfrac{3}{4}\text{ h}$, School B takes $\dfrac{1}{2}\text{ h}$, and School C takes $\dfrac{2}{5}\text{ h}$.

To solve this distribution with fractional coefficients, there are two common procedures:

• Method 1: Use the inverse fractions directly ($\dfrac{4}{3}$, $2$, and $\dfrac{5}{2}$), sum them by finding the common denominator to obtain the constant $k$, and multiply each fraction by $k$.

• Method 2: Pre-multiply all the inverses by the least common multiple of their denominators ($\text{lcm}(3, 2) = 6$) to transform them into equivalent integers, and then perform a distribution directly proportional to those integers.

1. a) Solve the distribution by applying Method 2 step by step and calculate how much money each school receives.
2. b) Formally justify why Method 2 is valid (that is, why multiplying all coefficients by a positive constant does not alter the proportions or the final money assigned to each school).
3. c) Explain with reasoning why, when solving problems without a calculator, Method 2 usually minimizes the probability of making calculation errors compared to Method 1.

The Provincial Council allocates an extraordinary fund of $98000$ € from the Forest Fire Prevention Plan among three municipalities ($A$, $B$, and $C$). The technical committee establishes a compound distribution:

• Directly proportional to the forest area of the municipality: municipality $A$ has $1200\text{ ha}$, $B$ has $1800\text{ ha}$, and $C$ has $3000\text{ ha}$.

• Inversely proportional to the number of professional firefighters previously hired by each town council (in order to compensate municipalities with fewer staff of their own): $A$ has $2$ firefighters, $B$ has $4$, and $C$ has $5$.

1. a) Model the problem: calculate the weighted coefficient of each municipality by combining both criteria and simplify the ratio to the smallest possible integers.
2. b) Find the proportionality constant and the financial amount corresponding to each municipality.
3. c) Interpret the result: municipality $C$ has $2,5$ times more forest area than municipality $A$; however, does it receive $2,5$ times more money? Explain mathematically how the staffing variable influences this territorial compensation.

A total amount of $74$ is divided inversely proportional to three consecutive positive integers: $x$, $x+1$, and $x+2$. Knowing that the constant of inverse proportionality of the division is $k = 120$:

1. a) Set up the corresponding equation and find the three integers.
2. b) Determine the value of each of the three shares obtained in the division.

A prize of $6000$ € is to be shared among three cyclists at the end of a grand tour. The rules state that the sharing will be directly proportional to the number of stages won ($3$, $4$ and $5$ stages) and inversely proportional to the number of crashes suffered ($1$, $2$ and $x$ crashes). If the cyclist who won $4$ stages and suffered $2$ crashes received exactly $2000$ €, determine the number of crashes $x$ of the third cyclist and the amount of money they received.

A triangular plot of land is divided into three parcels by drawing straight lines from an interior point $P$ to the three vertices. The areas of the three resulting parcels ($S_1$, $S_2$ and $S_3$) are directly proportional to the numbers $3$, $4$ and $5$. If the plot of land has a total area of $2400$ m$^2$ and is sold at a price of $15$ €/m$^2$, calculate the monetary value of each of the three parcels.

A grandmother wants to divide $4200$ € among her three grandchildren aged $10$, $12$ and $15$ years old. She hesitates between doing it directly proportional to their ages or inversely proportional to them. Calculate how much each grandchild would receive under both options and state which grandchild experiences the greatest absolute difference in money between the two methods.

An amount of $8700$ € has been divided inversely proportional to three unknown numbers $A$, $B$ and $C$. The amounts received were $4000$ €, $3000$ € and $1700$ € respectively. Knowing that the smallest of the three original numbers is $51$, determine the value of the other two numbers.

A technology company shares $50000$ € of profit among three departments in two phases. In the first phase, $40$% of the total is shared directly proportional to the overtime hours worked: $120$, $150$ and $130$ hours. In the second phase, the remaining $60$% is shared inversely proportional to the number of technical incidents reported: $2$, $4$ and $5$ incidents. Calculate the total final profit received by each of the three departments by combining both phases.

Divide the algebraic expression $N = 6x^2 + 8x$ directly proportional to the expressions $x$, $x+1$ and $x+3$. Express the three resulting parts in terms of $x$ and fully simplify the polynomials obtained.

Three different pipes fill an industrial tank. The first takes $a$ hours, the second $b$ hours and the third $c$ hours. It is known that the times $a$, $b$ and $c$ are directly proportional to the numbers $2$, $3$ and $4$. The maintenance costs of the pipes amount to $1300$ € and it is decided to split them inversely proportional to the time each one takes to fill the tank. Find how much maintenance must be paid for each pipe.

In a regional technological projects competition, a total prize of $3500$ € is distributed among the three finalist teams inversely proportional to the final position obtained: $1.^{\text{st}}$ place, $2.^{\text{nd}}$ place, and $n$-th place ($1$, $2$, and $n$, with $n \in \mathbb{N}$ and $n > 2$).

After applying the regulatory distribution, the organization notes that the team classified in first place has received exactly $1500$ € more than the team classified in $n$-th place.

1. a) Set up an equation in terms of the unknown $n$ using the algebraic expression of each share, the total distributed, and the difference between the first and the last placed teams.
2. b) Solve the equation and deduce with justification in which position ($n$) the third classified team finished.
3. c) Using the obtained value of $n$, calculate the amount assigned to the runner-up and rigorously verify that the sum of the three shares matches the total of $3500$ €.

An irrigation community in the Guadalquivir valley must distribute the monthly maintenance and solar pumping cost ($7800$ €) among three agricultural holdings ($A$, $B$, and $C$).

Initially, a distribution directly proportional to the volume of water consumed during the month was used: $A$ consumed $15000\text{ m}^3$, $B$ consumed $25000\text{ m}^3$, and $C$ consumed $20000\text{ m}^3$.

To encourage the efficient use of water, the assembly approves replacing it with a compound distribution: directly proportional to the volume consumed and inversely proportional to the efficiency index of the installed irrigation system ($E_A = 0,75$; $E_B = 0,50$; $E_C = 0,40$, where the indices are normalized between $0$ and $1$).

1. a) Model both scenarios: calculate how much each holding paid under the initial criterion and how much it pays under the efficiency compound distribution.
2. b) Justify which holding experiences the largest percentage increase compared to its initial bill and reason the mathematical cause of this increase.
3. c) Holding $C$ plans to modernize its technology so that, maintaining its consumption of $20000\text{ m}^3$ and assuming that $A$ and $B$ do not change their data, its monthly fee does not exceed $2600$ € (what it paid under the initial criterion). Set up an inequality with unknown $E_C$ and calculate the minimum efficiency index it must achieve. Is it technically viable considering that the maximum possible efficiency is $1$?